Feedback linearized trajectory-tracking control of a mobile robot

This paper is devoted to the designing of a trajectory-tracking control system for a unicycle-type mobile robot. Synthesis of the trajectory control law is based on the feedback linearization method and a canonical similarity transformation of nonlinear affine system in state-dependent coefficient form. The result of experimental test of the trajectory control system for mobile robot Rover5 is presented.


Introduction
The problem of mobile robot (MR) control is relevant over the years because of the wide range of theoretical problems and practical applications associated with it.There is a great number of works in the field of motion control of MR that are published in recent decades.Their reviews can be founded in [1][2][3][4].
Trajectory-tracking control problem is a typical control problem for MR.It is mainly concerned with the design of control laws that force a MR to reach and follow a time parameterized reference trajectory [5].Thus, as a rule, simple non-linear models are operated, which characterize only the kinematic relationship between the MR's motion parameters and parameters that are predetermined by a reference trajectory.
At this point it is possible to mark out two basic approaches to design the tra-jectory-tracking system.The first approach is based on linear control design methods and tangent linearization of system model about the reference trajecto-ry.At that, a pole placement method or linear optimal control method are usually used to adjust controller parameters [6][7][8].Another approaches is based on non-linear control design methods, such as feedback linearization (FL), Lyapunov techniques [9][10][11].
A lot of publications are devoted to the trajectorytracking control problem for MR with nonholonomic constraints by means of FL (static and dynamic).In [9] applicability of static FL for setpoint regulation problems and for trajectory-tracking problems is considered.In [11] the trajectory-tracking control problem based on dynamic FL is considered.It is shown that dynamic FL is an efficient design tool leading to a solution simultaneously valid for both trajectory tracking and setpoint regulation problems.The main drawback of such solution is aligned with the raise of the system order and the order of the dynamic feedback.
In this paper, we propose another approach for static FL control design for tra-jectory-tracking, based on representing the original nonlinear system into a state-dependent coefficient form and applying the canonical similarity transformation, that allow getting the system to canonical form.This similarity transformation allow accomplishing linearization of a system without determining of a virtual system output.
The remaining sections of the paper proceed as follows: Section 2 describes the trajectory-tracking control problem statement; Section 3 is devoted to the problem of designing of trajectory control system for MR based on FL method; Section 4 describes the structure of the trajectory control system, results of experimental tests for MR Rover5; some conclusions are shown in Section 5.

Formulation of a Trajectory-Tracking Control problem for MR
The simplest model of a nonholonomic MR is the unicycle.Let us consider the kinematic model of a unicycle-type MR motion in a horizontal plane (Fig. 1a) [1][2][3][4]: ), ( sin 2 where Y X , are coordinates of MR posture in the earth coordinate system; 1 are linear velocities of left and right wheels and velocity of MR; ω is a MR angular velocity; ϕ is an angle between the vector V and the axis OX .
We suppose that there is a given trajectory of MR MR trajectory-tracking errors (Fig. 1b) is defined as follows [1][2][3][4]  ., cos , , sin , The problem is to find control inputs 1 u and 2 u , such that system (3) be an asymptotically stable, i.e.The procedure of constructing the similarity transformation is given in detail in [13].Here is an example of the system conversion for the case 2 R u ∈ .
Applying the similarity transformation . Feedback linearizing control for system (4) can be defined in the next form . ), ( Instead of elements ij k , it is necessary to take corresponding elements of ., kl k E J

Design of Trajectory-Tracking Control Law
Trajectory-tracking control loop is designed on the basis of FL method (4), (5) for tracking error dynamic system (3), which can be written in a state-dependent coefficient form: . sin

Numerical values of
The control law ( 9) is not applicable when 0 =

The experimental results
Object of the study is a MR Rover5 by DAGU [8] , m.For measuring the motor's speed MR Rover5 is equipped with two quadrature encoders.Encoder's resolution is 1000 pulses for 3 turns of the driving wheel.
An eight-shaped reference trajectory was taken as a test trajectory.At the initial time the robot is at the point with coordinates (0,25; -0,25), the trajectory starts to move from the point (0,0).The experimental results on a real Rover5 robot are shown in Figure 2.

Conclusions
The problem of MR trajectory-tracking control was considered.The design of trajectory control loop is based on FL problem for tracking error dynamic.In general case, the control law is a tracking error feedback with time-varying coefficients.Non-stationary nature of the problem is determined by the time-varying values of reference linear and angular velocities of the robot.In such a case when these velocities are constant, the problem becomes stationary and the feedback's coefficients also become time-invariant.Experimental verification of developed trajectorytracking control system was performed on mobile robot Rover5.The results of experiment confirm the acceptable quality of designed control system.This work was supported by the Russian Science Foundation (project No. 17-11-01220)

Fig. 1 .
Fig. 1. a The MR on a plane.b Statement of the trajectorytracking control problem for MR.
means of which the system (4) can be transformed into a linear canonical form.

.
So on the basis of this law it is impossible to realize a tracking along a straightforward trajectory.To overcome this disadvantage, it is possible to change the similarity transformation:In this case, the resulting control law has the form(8

Fig. 2 .
Fig. 2. The experimental results: reference and real trajectory of MR Rover5.Reference and real Rover5's trajectories are shown on the fig.2.The required req V and real V linear velocities, required req ω and real ω robot's angular velocities are shown on the fig.3.The results show us that the system has an acceptable quality.Robot enters the reference trajectory for 5 seconds, then deviations are 015 .0 ≤ X e, m,

Fig. 3 .
Fig. 3.The experimental results: reference and real trajectory of MR Rover5.